Question

Suppose that you and a friend are playing cards and decide to make a bet. If you draw three fives in succession from a standard deck of 52 cards with replacement, you win $70. Otherwise, you pay your friend $10. What is the expected value of your bet? Round your answer to the nearest cent, if necessary.

Answer 1

[AI-generated answer — Bot GPT 3.5]

To calculate the expected value of the bet, we first need to determine the probability of drawing three fives in succession with replacement.

There are 52 cards in a standard deck, and there are 4 fives in the deck. The probability of drawing a five on the first draw is 4/52. Since we are replacing the card after each draw, the probability of drawing a five on the second draw is also 4/52, and the probability of drawing a five on the third draw is also 4/52.

Therefore, the probability of drawing three fives in succession is (4/52) * (4/52) * (4/52) = (1/13) * (1/13) * (1/13) = 1/2197.

If you win $70 when you draw three fives in succession and lose $10 otherwise, the expected value of your bet can be calculated as follows:

Expected value = (Probability of winning * Amount won) + (Probability of losing * Amount lost)
Expected value = (1/2197 * $70) + (2196/2197 * (-$10))
Expected value = $0.02 - $10
Expected value = -$9.98

Therefore, the expected value of the bet is -$9.98. This means that on average, you can expect to lose $9.98 every time you play this game.